Zur Vollständigkeit der induktiven Gruppoide der partiellen Automorphismen von Algebren. (On the completeness of the inductive groupoids of the partial automorphisms of algebras) (Q581660)

For any (total) algebra \({\mathcal A}\) with underlying set A, set inclusion on the powerset of \(A\times A\) reduces to an inductive partial order on the set \(\Gamma\) (\({\mathcal A})\) of partial automorphisms of \({\mathcal A}\) (isomorphisms between subalgebras), in the sense that every nonempty subset of \(\Gamma\) (\({\mathcal A})\) has an infimum. If binary infima distribute over suprema of sets with an upper bound, the partial order is called local. On the other hand \(\Gamma\) (\({\mathcal A})\) is a groupoid under the partial operation of composition, and this groupoid structure is compatible with the partial order, yielding an inductive groupoid in the sense of Ehresmann. The author's main concern is the following completeness property. Call a subset K of \(\Gamma\) (\({\mathcal A})\) compatible if the domain and codomain operation commute with binary infima on K. Then \(\Gamma\) (\({\mathcal A})\) is said to be complete if for every compatible subset K the existence of upper bounds for the sets of domains and codomains, respectively, of elements of K implies the existence of an upper bound (and hence a supremum) for K itself. Among other results, the author characterizes the finite groups G for which \(\Gamma\) (G) is complete, and shows that finite acyclic rings as well as finite fields have complete inductive groupoids of partial automorphisms.